4. Amplitude Ratio and Phase Shift of First Order System

For a first order system, \[ G(s) = \frac{K_p}{\tau_ps+1} \] Put s=j\( \omega \). \begin{align*} G(j\omega) &= \frac{K_p}{\tau_p j\omega +1} \\ &= \frac{K_p}{1+j\omega\tau_p} = \frac{K_p}{(1+j\omega\tau_p)}\frac{(1-j\omega\tau_p)}{(1-j\omega\tau_p)} \\ &= \frac{K_p}{\tau_p^2\omega^2+1}-j\frac{K_p\tau_p\omega}{\tau_p^2\omega^2+1} \tag*{(1)} \end{align*}

For Eqn.(1), \[\begin{aligned} \text{Modulus of } G(j\omega) &= \sqrt{\left(\frac{K_p}{\tau_p^2\omega^2+1}\right)^2+\left(\frac{-K_p\omega\tau_p}{\tau_p^2\omega^2+1}\right)^2} \\ &=\sqrt{\frac{K_p^2+K_p^2\omega^2\tau_p^2}{(\tau_p^2+1)^2}} \\ &= \frac{K_p}{\sqrt{\tau_p^2\omega^2+1}} = \text{ Amplitude Ratio} = \text{AR} \end{aligned}\]

For Eqn.(1), \[\begin{aligned} \text{Argument of } G(j\omega) &= \tan^{-1}(-\omega\tau_p) = \text{Phase Shift} = \phi \end{aligned}\] This is a phase lag since \(\phi<0\).